Electronic filter
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
defines
electronic filter
Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributed-element filters. That is, using components ...
circuits without taking note of the values of the components used but only the manner in which those components are connected.
Filter design
Filter design is the process of designing a signal processing filter that satisfies a set of requirements, some of which may be conflicting. The purpose is to find a realization of the filter that meets each of the requirements to a sufficient ...
characterises filter circuits primarily by their
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
rather than their
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. Transfer functions may be
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
or
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
. Common types of linear filter transfer function are;
high-pass
A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency de ...
,
low-pass
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
,
bandpass
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-port ...
,
band-reject or notch and
all-pass. Once the transfer function for a filter is chosen, the particular topology to implement such a
prototype filter
Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or tra ...
can be selected so that, for example, one might choose to design a
Butterworth filter
The Butterworth filter is a type of signal processing filter designed to have a frequency response that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the Br ...
using the
Sallen–Key topology.
Filter topologies may be divided into
passive
Passive may refer to:
* Passive voice, a grammatical voice common in many languages, see also Pseudopassive
* Passive language, a language from which an interpreter works
* Passivity (behavior), the condition of submitting to the influence of on ...
and
active
Active may refer to:
Music
* ''Active'' (album), a 1992 album by Casiopea
* Active Records, a record label
Ships
* ''Active'' (ship), several commercial ships by that name
* HMS ''Active'', the name of various ships of the British Royal ...
types. Passive topologies are composed exclusively of
passive components
Passivity is a property of engineering systems, most commonly encountered in analog electronics and control systems. Typically, analog designers use ''passivity'' to refer to incrementally passive components and systems, which are incapable of p ...
: resistors, capacitors, and inductors. Active topologies also include
active components (such as transistors, op amps, and other integrated circuits) that require power. Further, topologies may be implemented either in
unbalanced form or else in
balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
form when employed in
balanced circuit A balanced circuit is circuitry for use with a balanced line or the balanced line itself. Balanced lines are a common method of transmitting many types of electrical communication signals between two points on two wires. In a balanced line the tw ...
s. Implementations such as
electronic mixer
An electronic mixer is a device that combines two or more electrical or electronic signals into one or two composite output signals. There are two basic circuits that both use the term ''mixer'', but they are very different types of circuits: a ...
s and
stereo sound
Stereophonic sound, or more commonly stereo, is a method of sound reproduction that recreates a multi-directional, 3-dimensional audible perspective. This is usually achieved by using two independent audio channels through a configuration ...
may require arrays of identical circuits.
Passive topologies
Passive filters have been
long in development and use. Most are built from simple
two-port network
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
s called "sections". There is no formal definition of a section except that it must have at least one series component and one shunt component. Sections are invariably connected in a
"cascade" or
"daisy-chain" topology, consisting of additional copies of the same section or of completely different sections. The rules of series and parallel
impedance would combine two sections consisting only of series components or shunt components into a single section.
Some passive filters, consisting of only one or two filter sections, are given special names including the L-section, T-section and Π-section, which are unbalanced filters, and the C-section, H-section and box-section, which are balanced. All are built upon a very simple "ladder" topology (see below). The chart at the bottom of the page shows these various topologies in terms of general
constant k filter
Constant k filters, also k-type filters, are a type of electronic filter designed using the image method. They are the original and simplest filters produced by this methodology and consist of a ladder network of identical sections of passive com ...
s.
Filters designed using
network synthesis
Network synthesis is a design technique for linear electrical circuits. Synthesis starts from a prescribed impedance function of frequency or frequency response and then determines the possible networks that will produce the required response. ...
usually repeat the simplest form of L-section topology though component values may change in each section.
Image designed filters, on the other hand, keep the same basic component values from section to section though the topology may vary and tend to make use of more complex sections.
L-sections are never symmetrical but two L-sections back-to-back form a symmetrical topology and many other sections are symmetrical in form.
Ladder topologies
Ladder topology, often called Cauer topology after
Wilhelm Cauer
Wilhelm Cauer (24 June 1900 – 22 April 1945) was a German mathematician and scientist. He is most noted for his work on the analysis and synthesis of electrical filters and his work marked the beginning of the field of network synthesis. Prio ...
(inventor of the
elliptic filter
An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amo ...
), was in fact first used by
George Campbell (inventor of the
constant k filter
Constant k filters, also k-type filters, are a type of electronic filter designed using the image method. They are the original and simplest filters produced by this methodology and consist of a ladder network of identical sections of passive com ...
). Campbell published in 1922 but had clearly been using the topology for some time before this. Cauer first picked up on ladders (published 1926) inspired by the work of Foster (1924). There are two forms of basic ladder topologies; unbalanced and balanced. Cauer topology is usually thought of as an unbalanced ladder topology.
A ladder network consists of cascaded asymmetrical L-sections (unbalanced) or C-sections (balanced). In
low pass form the topology would consist of series inductors and shunt capacitors. Other bandforms would have an equally simple topology
transformed from the lowpass topology. The transformed network will have shunt admittances that are
dual networks of the series impedances if they were duals in the starting network - which is the case with series inductors and shunt capacitors.
Modified ladder topologies
Image filter design commonly uses modifications of the basic ladder topology. These topologies, invented by
Otto Zobel
Otto Julius Zobel (October 20, 1887 – January 1970) was an electrical engineer who worked for the American Telephone & Telegraph Company (AT&T) in the early part of the 20th century. Zobel's work on filter design was revolutionary and led ...
, have the same
passband
A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
s as the ladder on which they are based but their transfer functions are modified to improve some parameter such as
impedance matching
In electronics, impedance matching is the practice of designing or adjusting the input impedance or output impedance of an electrical device for a desired value. Often, the desired value is selected to maximize power transfer or minimize signal ...
,
stopband
A stopband is a band of frequencies, between specified limits, through which a circuit, such as a filter or telephone circuit, does not allow signals to pass, or the attenuation is above the required stopband attenuation level. Depending on applic ...
rejection or passband-to-stopband transition steepness. Usually the design applies some transform to a simple ladder topology: the resulting topology is ladder-like but no longer obeys the rule that shunt admittances are the dual network of series impedances: it invariably becomes more complex with higher component count. Such topologies include;
*
m-derived filter
m-derived filters or m-type filters are a type of electronic filter designed using the image method. They were invented by Otto Zobel in the early 1920s. This filter type was originally intended for use with telephone multiplexing and was a ...
*
mm'-type filter
mm'-type filters, also called double-m-derived filters, are a type of electronic filter designed using the image method. They were patented by Otto Zobel in 1932.Zobel, O J, ''Electrical wave filters'', US patent 1 850 146, filed 25 Nov 1930, issu ...
*
General mn-type filter
The m-type (m-derived) filter is by far the most commonly used modified image ladder topology. There are two m-type topologies for each of the basic ladder topologies; the series-derived and shunt-derived topologies. These have identical transfer functions to each other but different image impedances. Where a filter is being designed with more than one passband, the m-type topology will result in a filter where each passband has an analogous frequency-domain response. It is possible to generalise the m-type topology for filters with more than one passband using parameters m
1, m
2, m
3 etc., which are not equal to each other resulting in general m
n-type filters which have bandforms that can differ in different parts of the frequency spectrum.
The mm'-type topology can be thought of as a double m-type design. Like the m-type it has the same bandform but offers further improved transfer characteristics. It is, however, a rarely used design due to increased component count and complexity as well as its normally requiring basic ladder and m-type sections in the same filter for impedance matching reasons. It is normally only found in a
composite filter.
Bridged-T topologies
![Zobel Bridged-T Filter](https://upload.wikimedia.org/wikipedia/commons/3/34/Zobel_Bridged-T_Filter.svg)
Zobel constant resistance filters use a topology that is somewhat different from other filter types, distinguished by having a constant input resistance at all frequencies and in that they use resistive components in the design of their sections. The higher component and section count of these designs usually limits their use to equalisation applications. Topologies usually associated with constant resistance filters are the bridged-T and its variants, all described in the
Zobel network : ''For the wave filter invented by Zobel and sometimes named after him see m-derived filters.''
Zobel networks are a type of filter section based on the image-impedance design principle. They are named after Otto Zobel of Bell Labs, who publish ...
article;
*Bridged-T topology
*Balanced bridged-T topology
*Open-circuit L-section topology
*Short-circuit L-section topology
*Balanced open-circuit C-section topology
*Balanced short-circuit C-section topology
The bridged-T topology is also used in sections intended to produce a signal delay but in this case no resistive components are used in the design.
Lattice topology
![Lattice All-pass Phase Correction Filter](https://upload.wikimedia.org/wikipedia/commons/b/b2/Lattice_All-pass_Phase_Correction_Filter.svg)
Both the T-section (from ladder topology) and the bridge-T (from Zobel topology) can be transformed into a lattice topology filter section but in both cases this results in high component count and complexity. The most common application of lattice filters (X-sections) is in
all-pass filter
An all-pass filter is a signal processing filter that passes all frequencies equally in gain, but changes the phase relationship among various frequencies. Most types of filter reduce the amplitude (i.e. the magnitude) of the signal applied to i ...
s used for
phase equalisation.
Although T and bridged-T sections can always be transformed into X-sections the reverse is not always possible because of the possibility of negative values of inductance and capacitance arising in the transform.
Lattice topology is identical to the more familiar bridge topology, the difference being merely the drawn representation on the page rather than any real difference in topology, circuitry or function.
Active topologies
Multiple feedback topology
Multiple feedback topology is an electronic filter topology which is used to implement an
electronic filter
Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributed-element filters. That is, using components ...
by adding two poles to the
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
. A diagram of the circuit topology for a second order low pass filter is shown in the figure on the right.
The transfer function of the multiple feedback topology circuit, like all second-order
linear filter
Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using ...
s, is:
:
.
In an MF filter,
:
:
:
:
is the
Q factor
In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...
.
:
is the DC voltage
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
:
is the corner frequency
For finding suitable component values to achieve the desired filter properties, a similar approach can be followed as in the
Design choices section of the alternative Sallen–Key topology.
Biquad filter topology
''For the digital implementation of a biquad filter, see
Digital biquad filter
In signal processing, a digital biquad filter is a second order recursive linear filter, containing two poles and two zeros. "Biquad" is an abbreviation of "''biquadratic''", which refers to the fact that in the Z domain, its transfer function i ...
.''
A biquad filter is a type of
linear filter
Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using ...
that implements a
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
that is the ratio of two
quadratic function
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
s. The name ''biquad'' is short for ''
biquadratic
In algebra, a quartic function is a function of the form
:f(x)=ax^4+bx^3+cx^2+dx+e,
where ''a'' is nonzero,
which is defined by a polynomial of degree four, called a quartic polynomial.
A ''quartic equation'', or equation of the fourth degre ...
''. Any second-order filter topology can be referred to as a ''biquad'', such as the MFB or Sallen-Key.
However, there is also a specific "biquad" topology. It is also sometimes called the 'ring of 3' circuit.
Biquad filters are typically
active
Active may refer to:
Music
* ''Active'' (album), a 1992 album by Casiopea
* Active Records, a record label
Ships
* ''Active'' (ship), several commercial ships by that name
* HMS ''Active'', the name of various ships of the British Royal ...
and implemented with a single-amplifier biquad (SAB) or two-integrator-loop topology.
* The SAB topology uses feedback to generate
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
s and possibly complex
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
s. In particular, the feedback moves the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
poles of an
RC circuit
A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC ci ...
in order to generate the proper filter characteristics.
* The two-integrator-loop topology is derived from rearranging a biquadratic transfer function. The rearrangement will equate one signal with the sum of another signal, its integral, and the integral's integral. In other words, the rearrangement reveals a
state variable filter
A state variable filter is a type of active filter in electronic circuits. It consists of one or more integrators, connected in some feedback configuration. It is essentially used when precise Q factor is required, as other multi-order filters are ...
structure. By using different states as outputs, any kind of second-order filter can be implemented.
The SAB topology is sensitive to component choice and can be more difficult to adjust. Hence, usually the term biquad refers to the two-integrator-loop state variable filter topology.
Tow-Thomas filter
For example, the basic configuration in Figure 1 can be used as either a
low-pass
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
or
bandpass
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-port ...
filter depending on where the output signal is taken from.
The second-order low-pass transfer function is given by
:
where low-pass gain
. The second-order bandpass transfer function is given by
:
.
with bandpass gain
. In both cases, the
*
Natural frequency
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force.
The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all par ...
is
.
*
Quality factor
In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...
is
.
The bandwidth is approximated by
, and Q is sometimes expressed as a
damping constant . If a noninverting low-pass filter is required, the output can be taken at the output of the second
operational amplifier
An operational amplifier (often op amp or opamp) is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to c ...
, after the order of the second integrator and the inverter has been switched. If a noninverting bandpass filter is required, the order of the second integrator and the inverter can be switched, and the output taken at the output of the inverter's operational amplifier.
Akerberg-Mossberg filter
Figure 2 shows a variant of the Tow-Thomas topology, known as
Akerberg-Mossberg topology, that uses an actively compensated Miller integrator, which improves filter performance.
Sallen–Key topology
![Sallen-Key_Generic_Circuit](https://upload.wikimedia.org/wikipedia/commons/e/e3/Sallen-Key_Generic_Circuit.svg)
The Sallen-Key design is a non-inverting second-order filter with the option of high Q and passband gain.
See also
*
Prototype filter
Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or tra ...
*
Topology (electronics)
The topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with t ...
*
Linear filter
Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using ...
*
State variable filter
A state variable filter is a type of active filter in electronic circuits. It consists of one or more integrators, connected in some feedback configuration. It is essentially used when precise Q factor is required, as other multi-order filters are ...
Notes
References
*Campbell, G A, "Physical Theory of the Electric Wave-Filter", ''Bell System Technical Journal'', November 1922, vol. 1, no. 2, pp. 1–32.
*Zobel, O J, "Theory and Design of Uniform and Composite Electric Wave Filters", ''Bell System Technical Journal'', Vol. 2 (1923).
*Foster, R M, "A reactance theorem", ''Bell System Technical Journal'', Vol. 3, pp. 259–267, 1924.
*Cauer, W, "Die Verwirklichung der Wechselstromwiderstande vorgeschriebener Frequenzabhängigkeit", ''Archiv für Elektrotechnik'', 17, pp. 355–388, 1926.
*Zobel, O J, "Distortion correction in electrical networks with constant resistance recurrent networks", ''Bell System Technical Journal'', Vol. 7 (1928), p. 438.
*Zobel, O J, ''Phase-shifting network'', US patent 1 792 523, filed 12 March 1927, issued 17 Feb 1931.
External links
*{{Commonscatinline